Most of the results known for splines in the polynomial case extend in a natural way to the Tchebycheffian setting.
In particular, under suitable assumptions on the involved ECT-spaces, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), with similar properties to polynomial B-splines.
TB-splines were introduced in 1968 by Karlin [1]} using generalized divided differences.
There are several other ways to define them, including Hermite interpolation [2]}, [3]}, de Boor-like recurrence relations [4]}, [5]}, integral recurrence relations [6]}, and blossoming [7]}; see also the historical notes in [8]} for further details. Each of these definitions has advantages according to the properties to be proved and lead to the same functions, up to a proper scaling.
